Algorithm Design

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Chapter 9 PSPACE: A Class of Problems beyond NP
Since p(n) is a polynomial, so is n-p(n), and hence our space usage is
polynomial in n, as desired.
In summary, we have shown the following.
(9.4) QSAT can be solved in polynomial space. _
Extensions: An Algorithm for Competitive Facility Location
We can determine which player has a forced win in a game such as Competitive
Facility Location by a very similar type of algorithm.
Suppose Player 1 moves first. We consider all of his possible moves in
sequence. For each of these moves, we see who has a forced win in the resulting
game, with Player 2 moving first. If Player 1 has a forced win in any of them,
then Player 1 has a forced win from the initial position. The crucial point,
as in the QSAT algorithm, is that we can reuse the space from one candidate
move to the next; we need only store the single bit representing the outcome.
In this way, we only consume a polynomial amount of space plus th~ space
requirement for one recursive call on a graph with fewer nodes. As in the case
of QSAT, we get the recurrence
S(n) 1, O i requires C] for all j < i as prerequisites. When invoked, it
adds Q and deletes Cj for all j < i.
Now we ask: Is there a sequence of operators that will take us from eo = ¢ to
e* = {C1, C 2 ..... Cn}?
We claim the following, by induction on i:
From any configuration that does not contain C 1 for any j < i, there exists
a sequence of operators that reaches a configuration.containing C i for all
] < i; but any such sequence has at least 2 ~ - 1 steps.
This is clearly true for i = 1. For larger i, here’s one solution.
o By induction, achieve conditions {Q_I .....Q} using operators O1 .....
o Now invoke operator (9i, adding Ci but deleting everything else.
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